Supercongruences involving Motzkin numbers and central trinomial coefficients

Let M n and T n denote the n th Motzkin number and the n th central trinomial coefficient respectively. We prove that for any prime p≥ 5 , {equation*} {aligned}\\[-22pt] &∑ₖ₌₀ᵖ⁻¹M_k^2≡ (p/3)(2-6p){p^2},\\ &∑ₖ₌₀ᵖ⁻¹kM_k^2≡ (p/3)(9p-1){p^2},\\ &∑ₖ₌₀ᵖ⁻¹T_kM_k≡ 4/3(p/3)+p/6(1-9(p/3)){p^2},\\[-6pt] {aligned} {equation*} where (-) is the Legendre symbol. These results confirm three supercongruences conjectured by Z.-W. Sun in 2010.